ó <Zc@s‘dZddlZddlmZmZmZmZmZddlm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9ddl:m;Z;m<Z<m=Z=m>Z>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRe efZSeefZTe!efZUe"efZVe#e$fZWe#e$e%fZXd„ZYd„ZZd „Z[e\ed „Z]d „Z^ed „Z_d „Z`ed„Zaed„Zbded„Zcded„Zded„Zeefed„Zgd„Zhed„Zied„Zjed„Zked„Zled„Zmed„Zned„Zoed„Zped„Zqed„Zred „Zsed!„Zted"„Zued#„Zved$„Zwd%„Zxed&„Zyed'„Zzed(„Z{ed)„Z|d*„Z}ed+„Z~ed,„Zed-„Z€ed.„Zed/„Z‚ed0„Zƒed1„Z„ed2„Z…ed3„Z†ed4„Z‡ed5„Zˆed6„Z‰ed7„ZŠed8„Z‹ed9„ZŒed:ƒZed;ƒZŽd<„Zed=„Zed>„Z‘ed?„Z’ed@„Z“edA„Z”edB„Z•dCdD„Z–dCdE„Z—dCdF„Z˜dCdG„Z™edHkry+ddlšj›jœjZžežj€Z€ežj|Z|WneŸe fk r‰dIGHnXndS(Js- Low-level functions for complex arithmetic. iÿÿÿÿNi(tMPZtMPZ_ZEROtMPZ_ONEtMPZ_TWOtBACKEND(1t round_floort round_ceilingt round_downtround_upt round_nearestt round_fasttbitcounttbctablet normalizet normalize1treciprocal_rndtrshifttlshiftt giant_stepst negative_rndtto_strtto_fixedt from_man_expt from_floattto_floattfrom_inttto_inttfzerotfonetftwotfhalftfinftfninftfnantfnonetmpf_abstmpf_postmpf_negtmpf_addtmpf_subtmpf_multmpf_divt mpf_mul_intt mpf_shifttmpf_sqrtt mpf_hypott mpf_rdiv_intt mpf_floortmpf_ceiltmpf_ninttmpf_fractmpf_signtmpf_hasht ComplexResult(tmpf_pitmpf_exptmpf_logt mpf_cos_sint mpf_cosh_sinhtmpf_tant mpf_pow_intt mpf_log_hypottmpf_cos_sin_pitmpf_phitmpf_costmpf_sint mpf_cos_pit mpf_sin_pitmpf_atant mpf_atan2tmpf_coshtmpf_sinhtmpf_tanhtmpf_asintmpf_acost mpf_acosht mpf_nthroott mpf_fibonaccicCs0|\}}|tkrtS|tkr,tStS(s2Check if either real or imaginary part is infinite(t_infstTruetFalse(tztretim((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_is_inf)s    cCs0|\}}|tkrtS|tkr,tStS(s9Check if either real or imaginary part is infinite or nan(t _infs_nanRORP(RQRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_is_infnan0s    cKsg|\}}t||ƒ}|drG|dtt|ƒ||dS|dt|||dSdS(Nis - tjs + (RR%(RQtdpstkwargsRRRStrs((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_to_str7s   "cCs1|\}}tt|||ƒt|||ƒƒS(N(tcomplexR(RQtstricttrndRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_to_complex?s cCs’tjdkrY|\}}t|ƒtjjt|ƒ}|dtjj}t|ƒSytt|dt ƒƒSWnt k rt|ƒSXdS(Ns3.2iR]( tsystversionR4t hash_infotimagtwidthtintthashR_ROt OverflowError(RQRRRSth((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_hashCs    cCs"|\}}|t|||ƒfS(N(R%(RQtprecR^RRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_conjugatePs cCs |tkS(N(tmpc_zero(RQ((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_is_nonzeroTscCs@|\}}|\}}t||||ƒt||||ƒfS(N(R&(RQtwRjR^tatbtctd((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_addWs  cCs%|\}}t||||ƒ|fS(N(R&(RQtxRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_add_mpf\s icCs@|\}}|\}}t||||ƒt||||ƒfS(N(R'(RQRnRjR^RoRpRqRr((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_sub`s  cCs%|\}}t||||ƒ|fS(N(R'(RQtpRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_sub_mpfes cCs.|\}}t|||ƒt|||ƒfS(N(R$(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_posis cCs.|\}}t|||ƒt|||ƒfS(N(R%(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_negms cCs(|\}}t||ƒt||ƒfS(N(R+(RQtnRoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_shiftqs cCs|\}}t||||ƒS(sEAbsolute value of a complex number, |a+bi|. Returns an mpf value.(R-(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_absus cCs|\}}t||||ƒS(s3Argument of a complex number. Returns an mpf value.(RE(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_arg{s cCs.|\}}t|||ƒt|||ƒfS(N(R/(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_floor€s cCs.|\}}t|||ƒt|||ƒfS(N(R0(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_ceil„s cCs.|\}}t|||ƒt|||ƒfS(N(R1(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_nintˆs cCs.|\}}t|||ƒt|||ƒfS(N(R2(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_fracŒs cCsˆ|\}}|\}}t||ƒ}t||ƒ} t||ƒ} t||ƒ} t|| ||ƒ} t| | ||ƒ} | | fS(sÎ Complex multiplication. Returns the real and imaginary part of (a+bi)*(c+di), rounded to the specified precision. The rounding mode applies to the real and imaginary parts separately. (R(R'R&(RQRnRjR^RoRpRqRrRwtqtrtsRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_mul‘s  c Csm|\}}t||ƒ}t||ƒ}t||||ƒ}t||||ƒ}t|dƒ} || fS(Ni(R(R'R+( RQRjR^RoRpRwRƒR„RRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_square£s cCs@|\}}t||||ƒ}t||||ƒ}||fS(N(R((RQRwRjR^RoRpRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_mul_mpf­s cCsF|\}}tt||||ƒƒ}t||||ƒ}||fS(sB Multiply the mpc value z by I*x where x is an mpf value. (R%R((RQRtRjR^RoRpRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_mul_imag_mpf³s cCs@|\}}t||||ƒ}t||||ƒ}||fS(N(R*(RQR{RjR^RoRpRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_mul_int¼s c Cs¶|\}}|\}}|d}tt||ƒt||ƒ|ƒ} tt||ƒt||ƒ|ƒ} tt||ƒt||ƒ|ƒ} t| | ||ƒt| | ||ƒfS(Ni (R&R(R'R)( RQRnRjR^RoRpRqRrtwptmagtttu((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_divÂs   $$$cCs@|\}}t||||ƒ}t||||ƒ}||fS(sCalculate z/p where p is real(R)(RQRwRjR^RoRpRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_div_mpfÍs cCsn|\}}tt||ƒt||ƒ|dƒ}t||||ƒ}tt||||ƒƒ}||fS(sCalculate 1/z efficientlyi (R&R(R)R%(RQRjR^RoRptmRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_reciprocalÔs  (c Cs€|\}}tt||ƒt||ƒ|dƒ}tt||ƒ|||ƒ}ttt||ƒƒ|||ƒ}||fS(s)Calculate p/z where p is real efficientlyi (R&R(R)R%( RwRQRjR^RoRpR‘RRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_mpf_divÜs  ($cCsŠd}d}xq|r|d@rQ||||||||}}|d8}n||||d||}}|d}qW||fS(sgComplex integer power: computes (a+b*I)**n exactly for nonnegative n (a and b must be Python ints).iii((RoRpR{twretwim((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytcomplex_int_powäs  % !cCsT|dtkr't||d||ƒSttt||dƒ||dƒ||ƒS(Niii (Rt mpc_pow_mpftmpc_expR†tmpc_log(RQRnRjR^((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_powñsc Cs¤|\}}}}|dkr=t|d|||>||ƒS|dkrwt||dƒ}t|d||||ƒSttt||dƒ||dƒ||ƒS(Niiÿÿÿÿi (t mpc_pow_inttmpc_sqrtR˜RˆR™( RQRwRjR^tpsigntpmantpexptpbctsqrtz((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyR—ös  cCsŽ|\}}|tkr1t||||ƒtfS|tkrÃt||||ƒ}|d;}|dkrr|tfS|dkrˆt|fS|dkr¤t|ƒtfS|dkrÃtt|ƒfSn|dkrÓtS|dkrït|||ƒS|dkr t|||ƒS|dkr't|||ƒS|dkrTtt|| |dƒ||ƒS|\}}} } |\} } } }|rˆ| }n| r˜| } n| | }t|ƒ}||t | |ƒ}|dkra|dkrð||K}| } n| | K} | } t || |ƒ\}}t |t || ƒ||ƒ}t |t || ƒ||ƒ}||fSt tt||dƒ||dƒ||ƒS( Niiiiiiÿÿÿÿi'i (RR<R%tmpc_oneRyR‡R’R›tabstmaxR–RReR˜RŠR™(RQR{RjR^RoRptvtasigntamantaexptabctbsigntbmantbexptbbctdetabs_det exact_sizeRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyR›ÿsX               !          c Cs©|\}}|tkry|tkr.||fS|drZtt|ƒ||ƒ}t|fSt|||ƒ}|tfSn|d}|dstt||f|ƒ||ƒ}t|dƒ} t| ||ƒ}t|dƒ} t| |ƒ} t|| ||ƒ}nštt||f|ƒ||ƒ}t|dƒ} t| ||ƒ}t|dƒ} t| |ƒ} t|| ||ƒ}|drŸt|ƒ}t|ƒ}n||fS(s¥Complex square root (principal branch). We have sqrt(a+bi) = sqrt((r+a)/2) + b/sqrt(2*(r+a))*i where r = abs(a+bi), when a+bi is not a negative real number.iiiÿÿÿÿi(RR,R%R&R}R+R)R'( RQRjR^RoRpRSRRR‹RRŽR¥Rn((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyRœ's6         !!  cCs…d}tt||||ƒƒ}tt||||ƒƒ}yP|d|d|}|j}|j} tt|ƒƒ}tt| ƒƒ} Wn‰tk rt||ƒ}t||ƒ}t|ƒ} td| |ƒ} t||f| t f|ƒ\}} t |ƒ}t | ƒ} nXd} |} |}xJt ||| ƒD]5}t || |dƒ\}}t||d| ||ƒ}t||d| ||ƒ}||||||?}t|||ƒ}t|||ƒ}|||||?}| ||||?}||>|}||>|}||dt ||| ƒ|}||dt | || ƒ|} |} qBW|| fS(Ni2yð?gð?ii (ReRtrealRcRRgRR.RšRRRR–R(RoRpR{Rjtstartta1tb1R„RRRStfntnthtextratprevptextra1Rwtre2tim2tr4taptbptrectimctrebtimb((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_nthroot_fixedKsF    $ ## cCs|\}}|ddkrG|tkrGt||||ƒ}|tfS|dkrá|dkrctS|dkr…t||f||ƒS|dkrªtt||f||ƒSt||f| |dt|ƒ}tt|||ƒS|dkrêtd|dƒ}|\} } } } |\} }}}t||f|ƒ}|d |dd krê|d |d|krêt ||ƒ}t ||ƒ}t ||||ƒ\}}d}t || |||ƒ}t || |||ƒ}||fSnt |ƒ}|dd}t d||ƒ}t||f|tf||ƒ\}}t|d|d|d|d ||ƒ}t|d|d|d|d ||ƒ}||fS( su Complex n-th root. Use Newton method as in the real case when it is faster, otherwise use z**(1/n) iiiiÿÿÿÿiig333333ó?i iþÿÿÿiöÿÿÿi(RRLR¢RyRt mpc_nthrootRReR}RRÃRRR.RšR (RQR{RjR^RoRpRRtinversetprec2R¦R§R¨R©RªR«R¬R­tpftaftbfRSR·RµR¶((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyRÄrsB      $ 0  '++cCst|d||ƒS(s Complex cubic root. i(RÄ(RQRjR^((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_cbrt›sc Cs°|\}}|tkr(t|||ƒS|tkrJt|||ƒtfSt||d|ƒ}t||d|ƒ\}}t||||ƒ}t||||ƒ} || fS(sv Complex exponential function. We use the direct formula exp(a+bi) = exp(a) * (cos(b) + sin(b)*i) for the computation. This formula is very nice because it is pefectly stable; since we just do real multiplications, the only numerical errors that can creep in are single-ulp rounding errors. The formula is efficient since mpmath's real exp is quite fast and since we can compute cos and sin simultaneously. It is no problem if a and b are large; if the implementations of exp/cos/sin are accurate and efficient for all real numbers, then so is this function for all complex numbers. i(RR9R7R(( RQRjR^RoRpRŒRqR…RRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyR˜¡s   cCs9t|d|d||ƒ}t|||ƒ}||fS(Nii(R=R~(RQRjR^RRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyR™¼sc Cs¾|\}}|tkr.t|||ƒtfS|tkrPt|||ƒtfS|d}t||ƒ\}}t||ƒ\}} t||||ƒ} t|| ||ƒ} | t| ƒfS(sSComplex cosine. The formula used is cos(a+bi) = cos(a)*cosh(b) - sin(a)*sinh(b)*i. The same comments apply as for the complex exp: only real multiplications are pewrormed, so no cancellation errors are possible. The formula is also efficient since we can compute both pairs (cos, sin) and (cosh, sinh) in single stwps.i(RR@RFR9R:R(R%( RQRjR^RoRpR‹RqR…tchtshRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_cosÁs    c Cs¸|\}}|tkr.t|||ƒtfS|tkrPtt|||ƒfS|d}t||ƒ\}}t||ƒ\}} t||||ƒ} t|| ||ƒ} | | fS(sƒComplex sine. We have sin(a+bi) = sin(a)*cosh(b) + cos(a)*sinh(b)*i. See the docstring for mpc_cos for additional comments.i(RRARGR9R:R(( RQRjR^RoRpR‹RqR…RËRÌRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_sinÕs    cCs |\}}|\}}}}|\} } } } |tkrRt|||ƒtfS|tkrttt|||ƒfS|d} t|dƒ}t|dƒ}t|| ƒ\}}t|| ƒ\}}t||| ƒ}t||||ƒ}t||||ƒ}||fS(scComplex tangent. Computed as tan(a+bi) = sin(2a)/M + sinh(2b)/M*i where M = cos(2a) + cosh(2b).ii(RR;RHR+R9R:R&R)(RQRjR^RoRpR¦R§R¨R©RªR«R¬R­R‹RqR…RËRÌRŒRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_tanås     c CsÞ|\}}|tkr.t|||ƒtfSt|t|dƒ|dƒ}|tkrpt|||ƒtfS|d}t||ƒ\}}t||ƒ\}} t||||ƒ} t|| ||ƒ} | t| ƒfS(Nii(RRBR(R6RFR>R:R%( RQRjR^RoRpR‹RqR…RËRÌRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_cos_piøs     c CsØ|\}}|tkr.t|||ƒtfSt|t|dƒ|dƒ}|tkrptt|||ƒfS|d}t||ƒ\}}t||ƒ\}} t||||ƒ} t|| ||ƒ} | | fS(Nii(RRCR(R6RGR>R:( RQRjR^RoRpR‹RqR…RËRÌRRRS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_sin_pis     cCs$|\}}|tkrFt|||ƒ\}}|tft|ffS|tkr€t|||ƒ\}}|tf|tffS|d} t|| ƒ\}}t|| ƒ\}}t||||ƒ} t||||ƒ} t||||ƒ} t||||ƒ} | t| ƒf| | ffS(Ni(RR:R9R(R%(RQRjR^RoRpRËRÌRqR…R‹tcretcimtsretsim((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_cos_sins    cCsD|\}}|tkrFt|||ƒ\}}|tf|tffSt|t|dƒ|dƒ}|tkr t|||ƒ\}}|tft|ffS|d} t|| ƒ\}}t|| ƒ\}}t||||ƒ} t||||ƒ} t||||ƒ} t||||ƒ} | t| ƒf| | ffS(Nii(RR>R(R6R:R%(RQRjR^RoRpRqR…RËRÌR‹RÒRÓRÔRÕ((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_cos_sin_pi%s      cCs(|\}}t|t|ƒf||ƒS(s:Complex hyperbolic cosine. Computed as cosh(z) = cos(z*i).(RÍR%(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_cosh7s cCs4|\}}t||f||ƒ\}}||fS(s;Complex hyperbolic sine. Computed as sinh(z) = -i*sin(z*i).(RÎ(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_sinh<s cCs4|\}}t||f||ƒ\}}||fS(s>Complex hyperbolic tangent. Computed as tanh(z) = -i*tan(z*i).(RÏ(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_tanhBs c CsÜ|\}}|d}tt||ƒt|ƒf}tt||ƒ|f}t||ƒ}t||ƒ} t|| ||ƒ\}}tt|dƒƒt|dƒf} | dtkrØt|ƒrØ| dt f} n| S(Niiÿÿÿÿii( R&RR%R'R™RvR+R!RTR( RQRjR^RoRpR‹Rttytl1tl2R¥((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_atanIs  $gû:pΈä?gø?cCsù|\}}|d}|tkr;ttt|ƒ|ƒ}|ds|dkrft|||ƒtfSt|||ƒtfSq;|drèt||ƒ}tt|ƒ||ƒ} |dkrÌ|t| ƒfStt |dƒƒ| fSq;t|||ƒ} |dkrt| fSt||ƒ}t |dƒt| ƒfSnd} } |drdt|ƒ}d} n|drƒt|ƒ}d} ntt||ƒ}t t||ƒ} t | ||ƒ} t |||ƒ}t t | ||ƒdƒ}t |||ƒ}t |||ƒ}tt||ƒdsP|dkr>t||ƒ}qÔt||ƒ}n„t |||ƒ}|dst |t | | |ƒ|ƒ} t |||ƒ}t t |t | ||ƒ|ƒdƒ}|dkrött t||ƒ||ƒ|ƒ}qÔtt |t||ƒ|ƒ|ƒ}n·t |t | | |ƒ|ƒ} t |t|||ƒ|ƒ}t t | ||ƒdƒ}t |t||ƒ|ƒ}|dkr¹tt |||ƒ|ƒ}ntt |||ƒ|ƒ}tt||ƒdsØt |t | | |ƒ|ƒ}t|ƒdrZt |||ƒ}t |||ƒ}t t |||ƒdƒ}n-t|||ƒ}t t |||ƒdƒ}t |t |t|ƒ|ƒ}tt tt |t||ƒ|ƒ|ƒ|ƒ}nBttt |||ƒt|ƒ|ƒ}tt |||ƒ|ƒ}| rV|dkrGtt|ƒ||ƒ}qVt|ƒ}n| rx|dkrxt|ƒ}n| r™|dkr™t|ƒ}nt|d|d|d|d||ƒ}t|d|d|d|d||ƒ}||fS(s& complex acos for n = 0, asin for n = 1 The algorithm is described in T.E. Hull, T.F. Fairgrieve and P.T.P. Tang 'Implementing the Complex Arcsine and Arcosine Functions using Exception Handling', ACM Trans. on Math. Software Vol. 23 (1997), p299 The complex acos and asin can be defined as acos(z) = acos(beta) - I*sign(a)* log(alpha + sqrt(alpha**2 -1)) asin(z) = asin(beta) + I*sign(a)* log(alpha + sqrt(alpha**2 -1)) where z = a + I*b alpha = (1/2)*(r + s); beta = (1/2)*(r - s) = a/alpha r = sqrt((a+1)**2 + y**2); s = sqrt((a-1)**2 + y**2) These expressions are rewritten in different ways in different regions, delimited by two crossovers alpha_crossover and beta_crossover, and by abs(a) <= 1, in order to improve the numerical accuracy. i iiÿÿÿÿiii(RR'RR#RJRIR6RKR%R+R&R-R)R(tbeta_crossoverRDR,talpha_crossoverR8R (RQRjR^R{RoRpR‹tamtpiRqR¦RªR½R„R…talphatbetatb2RRtAxRrtc1tc2tAm1RS((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt acos_asin_s–                  ' '' 3' ++cCst|||dƒS(Ni(Rê(RQRjR^((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_acosçscCst|||dƒS(Ni(Rê(RQRjR^((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_asinêscCs@|\}}t|t|ƒf||ƒ\}}t|ƒ|fS(N(RìR%(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_asinhís $cCsRt|||ƒ\}}|ds.|tkr>t|ƒ|fS|t|ƒfSdS(Ni(RëRR%(RQRjR^RoRp((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_acoshóscCsš|d}t|t|ƒ}tt||ƒ}t||ƒ}t||ƒ}tt|||ƒdƒ}|dtkr–t|ƒr–t|df}n|S(Niiÿÿÿÿii(RsR¢RvR™R|R!RTR(RQRjR^R‹RoRpR¥((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_atanhüs c Csú|\}}|tkr.t|||ƒtfStt|d|dƒt|d|dƒƒ}||d}t|ƒ}tt|dƒt|ƒ}t|tf||ƒ} t ||ƒ} t | | |ƒ} t | | |ƒ} t | |||ƒ} | S(Niiii( RRMR¤R£R?R&R+R"RšRÐRRvR( RQRjR^RRRStsizeR‹RoRpRŽR¥((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_fibonacci s  3 tfcCs t‚dS(N(R5(RtRjR^((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpf_expjscCs¶|\}}|tkr(t|||ƒS|tkrPtt|ƒ||ƒtfStt|ƒ|dƒ}t||dƒ\}}t||||ƒ}t||||ƒ}||fS(Ni (RR9R7R%R((RQRjR^RRRSteyRqR…((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pytmpc_expjs   cCs t‚dS(N(R5(RtRjR^((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpf_expjpi(sc Cs|\}}|tkr(t|||ƒS|\}}}}|d} |rd| td||ƒ7} nttt| ƒ|| ƒƒ}|tkr¤t|||ƒtfSt||dƒ} t||dƒ\} } t| | ||ƒ}t| | ||ƒ}||fS(Ni i(RR>R¤R%R(R6R7( RQRjR^RRRStsigntmantexptbcR‹RôRqR…((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyt mpc_expjpi+s    tsages&Warning: Sage imports in libmpc failed(¡t__doc__R`tbackendRRRRRtlibmpfRRRRR R R R R RRRRRRRRRRRRRRRRRRR R!R"R#R$R%R&R'R(R)R*R+R,R-R.R/R0R1R2R3R4R5t libelefunR6R7R8R9R:R;R<R=R>R?R@RARBRCRDRERFRGRHRIRJRKRLRMR¢Rltmpc_twotmpc_halfRNRURTRVR[RPR_RiRkRmRsRuRvRxRytNoneRzR|R}R~RR€RR‚R†R‡RˆR‰RŠRRR’R“R–RšR—R›RœRÃRÄRÊR˜R™RÍRÎRÏRÐRÑRÖR×RØRÙRÚRÞRßRàRêRëRìRíRîRïRñRóRõRöRûtsage.libs.mpmath.ext_libmptlibstmpmatht ext_libmpt_lbmpt ImportErrortAttributeError(((s2lib/python2.7/site-packages/mpmath/libmp/libmpc.pyts¢ (ÿ1š                            ( $ ' )                 ˆ